Contact your local government to ask about local code and fire ordinances that might relate to building a chimney for a basement wood stove. Confirm that it is acceptable to use prefabricated chimney pipe to build a capped, exterior metal chimney.
Drill a hole through the concrete block wall to receive the insulated wall thimble -- locate the hole so that the stove pipe can be routed to it easily and the chimney pipe can extend vertically to above the peak of the roof line. be sure to wear eye protection while drilling. Insert the insulated wall thimble in the wall and secure it in place according to the manufacturer's instructions.
Connect the stove vent pipe to the inside of the wall thimble, and secure it in place with sheet metal screws.
Connect the chimney pipe tee to the outside of the wall thimble, and secure it into place with sheet metal screws. Install the chimney pipe wall support below the tee, according to the manufacturer's directions, so that the tee rests firmly on the support.
Connect class A chimney pipe to the top of the tee and secure it into place with sheet metal screws. Extend the pipe up the wall, securing it in place with wall bands -- use a masonry drill and anchors to attach the bands to concrete block. Support the pipe with one wall band for every eight feet of pipe.
Continue to add sections of pipe, securing them in place with sheet metal screws, to extend the chimney at least two feet above the peak of the roof. Slide the roof support over the chimney, and attach the braces to the roof with wood screws -- attach the roof support to the chimney with sheet metal screws.
Place the round chimney cap onto the top of the chimney and secure it with sheet metal screws.
Contact a professional to test the draft of your chimney and wood stove to ensure proper ventilation for the health of your family.